High linearity superconducting radio frequency magnetic field detector

ABSTRACT

A superconducting quantum interference devices (SQUID) comprises a superconducting inductive loop with at least two Josephson junction, whereby a magnetic flux coupled into the inductive loop produces a modulated response up through radio frequencies. Series and parallel arrays of SQUIDs can increase the dynamic range, output, and linearity, while maintaining bandwidth. Several approaches to achieving a linear triangle-wave transfer function are presented, including harmonic superposition of SQUID cells, differential serial arrays with magnetic frustration, and a novel bi-SQUID cell comprised of a nonlinear Josephson inductance shunting the linear coupling inductance. Total harmonic distortion of less than −120 dB can be achieved in optimum cases.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 12/543,582, filed Aug. 18, 2009, now U.S. Pat. No. 8,179,133,which issued May 15, 2012, which claims benefit of priority from U.S.Provisional Patent Application No. 61/089,861, filed Aug. 18, 2008, theentirety of which are expressly incorporated herein by reference.

STATEMENT OF GOVERNMENT RIGHTS

This invention was made in part with government support under the USOffice of Naval Research CRDF Grant RUP1-1493-MO-05. The US governmenthas certain rights in this invention.

FIELD OF THE INVENTION

The present invention relates to the field of detectors and devices formagnetic fields, and more particularly to magnetic field detectors thatoperate at radio frequencies.

BACKGROUND OF THE INVENTION

The Superconducting Quantum Interference Device, or SQUID, is well knownas a sensitive detector of weak magnetic fields. As indicated in FIG. 1,a SQUID is comprised of a superconducting loop containing one or moreJosephson junctions (indicated by X in FIG. 1), and magnetic flux Φ isinductively coupled into the loop through a coupling inductor L. AJosephson junction is known to act as a lossless nonlinear inductancebelow its critical current I_(c), and also exhibits nonlinear resistanceabove I_(c). A one-junction SQUID (FIG. 1A) comprises a single junctionand a loop, and exhibits a nonlinear impedance which depends on the fluxΦ in a periodic manner, with periodicity Φ₀=h/2e=2.07 fT-m², where h isPlanck's constant and e is the charge on the electron. The one-junctionSQUID does not have a direct voltage readout since the junction isshunted by a lossless superconducting inductor, so it must be embeddedin a radio-frequency (RF) circuit for its impedance to be measured. Forthis reason, this structure is sometimes called an RF-SQUID, althoughthe flux Φ can be at any frequency down to DC. Another SQUID device isthe two-junction SQUID (FIG. 1B), which is generally operated with a DCbias current greater than the device critical currentI_(c)=I_(c1)+I_(c2) of the two constituent junctions. This then exhibitsa DC voltage output across the Josephson junctions, modulated by theflux Φ in a way that is again periodic in Φ₀ (FIG. 1C). Thistwo-junction SQUID was historically called the DC-SQUID, since itoperates down to DC frequencies, although it may alternatively operatewith flux modulation up to gigahertz radio frequencies. The DC SQUID ismuch more commonly used than the RF SQUID, so in general usage, the termSQUID commonly refers to two-junction SQUID. Such a SQUID may be usednot only for low-frequency magnetic field detectors, but also forradio-frequency amplifiers and active radio antennas if appropriateinductive inputs are used.

Both high-T_(c) and low-T_(c) superconductor based SQUID-amplifiers havebeen studied during the past ten years [7], [8], [9], [15], [16], [17].See also Hilbert, U.S. Pat. No. 4,585,999, “RF amplifier based on a DCSQUID”. However, the characteristics of the amplifiers are still farfrom desired performance values. Despite the fact that the noisetemperature T_(n)≈1-3 K [15] is reasonably low, the dynamic range(amplitude ratio) D=(T_(sat)/T_(n))^(1/2) of the amplifiers is stronglylimited by their saturation temperature T_(sat), which is as low as100-150° K. [7], [15], [16]. The other disadvantage of theSQUID-amplifiers is a narrow range of linearity of the transferfunction. Implementation of a flux-locked-loop operating mode cansubstantially increase dynamic range and linearity, but at the same timethe external feed-back loop will limit the maximum operation frequencyto a few tens of megahertz at best. Therefore an internal negativefeedback has been suggested in order to increase dynamic range and tolinearize the transfer function of such an amplifier [8], [9]. Howeverthis is very problematic given typical low values of the SQUID-amplifiergain [7], [15], [16] of 10-15 dB, since higher amplification gain isneeded to effectively achieve the negative feedback. What is needed is away to use arrays of SQUIDs to achieve both greater dynamic range andgreater linearity, without requiring such negative feedback. Linearityis particularly important for processing signals at radio frequencies,where a nonlinear transfer function can give rise to undesired harmonicsand intermodulation products.

One approach to overcoming the drawbacks of SQUID-amplifiers isassociated with multi-element Josephson structures and arrays, includingSuperconducting Quantum Interference Filters (SQIF). See Schopohl, U.S.Pat. No. 6,690,162, “Device for High Resolution Measurement of MagneticFields”; Oppenlander, U.S. Pat. No. 7,369,093, “Superconducting QuantumAntenna.” A SQIF is an array (parallel, series or parallel-series) of DCSQUIDs with an unconventional array structure [10-12]. The SQIF voltageresponse is characterized by a single sharp peak. Contrary to the usualSQUID, which shows unique properties due to the strict periodicity ofits structure, the unique properties of the SQIF result from just theopposite, an unconventional non-periodic array structure. SQIFs aretherefore a new development of an intelligent network of SQUIDs. SQIFscertainly offer an approach to achieving increased dynamic range, butthis approach does not offer a clear way to achieve linearization ofthese fundamentally nonlinear devices.

SUMMARY OF THE INVENTION

The present invention takes a novel approach in combining nonlinearJosephson junctions and SQUIDs in a way that cancels their mutualnonlinearities, to achieve an overall device where the totalnonlinearities have been significantly reduced or eliminated.

Recently, an approach to synthesis of multi-SQUID structures capable ofproviding high linearity voltage response has been reported by thepresent inventors [1,2]. The approach is based on formation of serialstructures which are able to provide a periodic triangular voltageresponse to a homogeneous magnetic field B with “spatial frequency”spectrum as follows:

$\begin{matrix}{{{V(B)} = {\sum\limits_{k}{A_{k}{\cos( {k\;\omega_{0}B} )}}}},{A_{k} = {A_{0}{\sum\limits_{k}\frac{\sin^{2}( {k\;\omega_{0}\Delta\;{B/2}} )}{( {k\;\omega_{0}\Delta\;{B/2}} )^{2}}}}},} & (1)\end{matrix}$

where 2ΔB is width of the triangular pulse with repetition cycleB_(T)=2π/ω₀. Such a triangular transfer function is quite linear forflux maintained on a single leg of the triangle, in contrast to the moretypical SQUID transfer function that is closer to a sinusoid. Thepresent invention discloses three novel approaches to achieving apiecewise linear triangle wave transfer function.

1) Harmonic Superposition

The first approach is to use interferometer (SQUID) cells providingharmonic (sinusoidal) voltage response with “spatial frequency”kω₀=(2π/Φ₀)·a_(k), where a_(k) is effective area of the interferometercell which belongs to the k-th group. In the case when the input signalis a control line current I (instead of B), the effective area should bereplaced by the mutual inductance M_(k), i.e., kω₀=(2π/Φ₀)·M_(k). FIG. 2shows a crude block diagram of an algorithmic flowchart to optimize thelinearity with this approach. The voltage for the series arraycorresponds to the sum of the contributions from the individual cells,as in Equation (1).

It is significant that the spectrum of the triangle wave voltageresponse with minimum period B_(T)=2ΔB contains only odd harmonics withamplitudes decreasing monotonically with harmonic number k as 1/k²:A(kω ₀)=A ₀ /k ² ,k=2n−1,n=1,2, . . .  (2)

2) Differential Magnetic Frustrated Arrays

The second way is to make a differential scheme, using serial arrays ofinterferometer cells biased by current I_(b)=I_(C), where I_(C) is theinterferometer critical current (the sum of the critical currents of thetwo junctions). The voltage response of such an array is characterizedby numerous harmonics with monotonically decreasing amplitudes:

$\begin{matrix}{{A( {n\;\omega_{0}} )} = {\frac{A_{0}}{n^{2} - 1}.}} & (3)\end{matrix}$

Apart from a few first harmonics, the amplitude decrease law is quiteclose to 1/n², as for the triangle wave A differential scheme of twoserial arrays, each periodic in Φ₀, but with a relative offset of Φ₀/2between them (known as “frustration”) causes the voltage output of thetwo arrays to be out of phase, which provides subtraction of all evenharmonics, and therefore the resulting response becomes very close tothe triangular one with B_(T)=ΔB. A differential output also exhibitspractical advantages in terms of avoidance of interference and noise.

FIG. 3 shows schematically the differential array structure and magneticbiasing of the arrays. One can further increase the linearity of thedifferential array circuit voltage response. For this purpose, we shouldadd to the array structure a few cells with sinusoidal responses. Thesecells are DC SQUIDs biased well above critical current (I_(b)>2I_(c)).These additional cells are to correct the initial spectral components in(3) in order to approach the desired spectrum (2). FIG. 1 shows adifferential array structure consisting of two series arrays of DCinterferometers with biasing I_(b)=I_(c), where I_(c) is the criticalcurrent of the interferometers. In one of the arrays each cell is biasedby magnetic flux Φ₀/2. Additional flux Φ₀/4 is applied to all the cellsto set the operating point in the center of the linear regime.

If the interferometer cells in both serial arrays are replaced byparallel SQUID arrays, a high-performance parallel-series differentialstructure results. In order to maximize the linearity in suchparallel-series structure, one should select parallel array parameters,(e.g. SQUID cell area distribution). In order to find an optimaldistribution of cell parameters, one can use an iterative algorithm tofind the problem solution, starting from some initial approximation.FIG. 2 shows a crude block diagram of an algorithmic flowchart tooptimize the linearity with this approach.

One aspect of the present inventions relates to such a differentialstructure. Considerable increase in the voltage response linearityresults from the use of parallel SQIFs [3] with this structure insteadof ordinary parallel arrays.

3) Modified SQUID Cells

In a third preferred embodiment, the basic SQUID cell itself ismodified. This novel modified SQUID, shown schematically in FIG. 4, iscalled the bi-SQUID. Here the linear coupling inductance of the SQUID ismodified by shunting with a nonlinear inductive element, a thirdJosephson junction below its critical current. Surprisingly, thisnonlinear element modifies the nonlinear transfer function of the SQUIDto produce a linear transfer function for appropriately chosenparameters. The nonlinear small-signal shunt inductance is given by theJosephson inductanceL _(J)=Φ₀/2π(Ic ₃ ² −I _(sh) ²)^(0.5),  (4)

-   -   where I_(c3) is the critical current of the shunt junction, and        I_(sh) is the current passing through the junction. The        effective loop inductance is the parallel combination of the        main inductance L and the Josephson inductance L_(J). The loop        comprising the additional junction and the main inductance forms        a single-junction SQUID, so that one may call this modified        SQUID a bi-SQUID. FIG. 4 also shows the voltage response of both        the conventional two-junction SQUID (dashed line) and the        bi-SQUID (solid line), for identical junctions J1 and J2 with        critical current I_(c)/2 and shunt junction critical current        I_(c3)=1.15 I_(c), with linear inductance L=Φ₀/2πI_(c) and bias        current I_(B)=2I_(c). This shows a triangular transfer function        where the triangle edges are quite straight, in contrast to the        conventional two-junction SQUID with a transfer function that is        closer to |sin(Φ_(e)/2πΦ₀)|. Thus, this bi-SQUID uses a        Josephson junction as a nonlinear inductive element which can        largely cancel the nonlinearity otherwise associated with the        SQUID transfer function.

It is noted that while the linearized SQUID cells and arrays of thepresent invention are generally periodic in magnetic flux with periodΦ₀, the periodicities in terms of current and magnetic field depend oneffective coupling loop mutual inductances and areas. Therefore, suchlinearized devices with different effective areas (and thusperiodicities) may be combined in series arrays to achieve a new devicewhereby the periodicity of the total voltage output is altered or eveneliminated, in much the same way that conventional SQUIDs are combinedto forms SQIFs with increased dynamic range in a single response nearzero flux. Such a SQIF-like device based on elements of the presentinvention would maintain the high linearity of its components, whilealso achieving an enhanced dynamic range and increased output voltageresponse.

It is therefore an object to provide a superconducting detector for amagnetic field, adapted to provide a linear detector output over a rangeof an applied magnetic field, and corresponding method, comprising: asuperconducting quantum interference device having a device output whichis non-linear with respect to variations of an applied magnetic flux,the applied magnetic flux being a function of at least the appliedmagnetic field; and at least one Josephson junction having a non-linearresponse, the superconducting quantum interference device and the atleast one Josephson junction being together configured and havingappropriate operational parameters such that the non-linear response ofthe at least one Josephson junction compensates the non-linear responseof the device output, such that over the range of the applied magneticfield, the detector output is substantially linear.

It is also an object to provide a method for linearizing an output of asuperconducting detector for a magnetic field, adapted to provide adetector output, comprising a superconducting quantum interferencedevice (SQUID) having a SQUID output which has a response pattern whichis periodic and substantially non-linear with respect to variations ofmagnetic flux, the magnetic flux being a function of magnetic field,comprising compensating the SQUID output with at least onesuperconducting element having a nonlinear impedance, wherein the SQUIDand the at least one superconducting element are appropriatelyconfigured and operated under such conditions to thereby linearize atleast a portion of a quarter of a period of the periodic andsubstantially non-linear response of the SQUID output, to produce thedetector output which is a substantially linear function of the magneticfield.

It is a further object to provide a magnetic field detector array, andcorresponding method, comprising a plurality of superconducting quantuminterference devices and a common output, each having an intrinsicresponse which is periodic and non-linear with respect to magnetic flux,the magnetic flux being a function of a magnetic field, the plurality ofsuperconducting quantum interference devices each having a respectiveeffective cell area, the respective effective cell areas of theplurality of superconducting quantum interference devices beingnon-uniformly distributed, the array being appropriately configured suchthat the each of the plurality of superconducting quantum interferencedevices provides a contribution to the common output, and wherein thecommon output is substantially linear with respect to changes inmagnetic field over a range of magnetic fields outside of a null field,for which no one of the plurality of superconducting quantuminterference devices is substantially linear.

The linear detector output is typically periodic in the magnetic field(that is, having a periodic non-monotonic variation with respect tochanges in magnetic field), having an ascending linear portion and adescending linear portion, corresponding to a triangle wave.

The linear detector input and/or output may comprise a radio-frequencysignal.

The superconducting quantum interference device may comprise at leastone superconducting loop, wherein the superconducting loop comprises atleast one inductor adapted to couple magnetic flux into thesuperconducting loop. The superconducting quantum interference devicemay further comprise at least two Josephson junctions, and, for examplemay be configured as a DC-SQUID. The at least one inductor may beshunted by a resistor and/or by the at least one Josephson junction,which is configured to act as a variable inductor. The at least oneJosephson junction may comprises a single Josephson junction whichdirectly shunts the at least one inductor, and thus be configured, forexample, as a bi-SQUID. The operating parameters of the single Josephsonjunction and superconducting quantum interference device may be togetherselected to increase the linearity of the linear detector output.

The at least one Josephson junction may comprise at least two additionalsuperconducting loops, each comprising at least one inductor and atleast two Josephson junctions, whereby each of the inductors is adaptedto couple magnetic flux into its respective superconducting loop, andeach superconducting loop has a respective loop output, the respectiveloop outputs and the superconducting quantum interference device outputbeing combined to provide the linear device output. The linear deviceoutput may receive contributions from the respective loop outputs of theat least two additional superconducting loops in parallel and/or inseries with the superconducting quantum interference device output. Theat least two additional superconducting loops may be substantiallymagnetically shielded by a superconducting ground plane. The magneticfield may be coupled to the at least two additional superconductingloops by inductive coupling from a superconducting control line.

A power of the detector output may be at least 6 dB higher than a powerof the magnetic field. Thus, the detector may serve as an amplifier. Theapplied magnetic field may comprise a radio-frequency magnetic signal,the detector further comprising an active antenna element having a powergain of at least 3 dB, configured to receive a radio frequency signaland present the radio frequency magnetic signal corresponding to theradio frequency signal.

The superconducting quantum interference device may comprise an inductorand at least two Josephson junctions, and wherein the at least oneJosephson junction comprises an additional superconducting loop,comprising at least one inductor and at least two Josephson junctionshaving an additional output, wherein the linear detector outputcomprises a differential output of the device output and the additionaloutput. The device output of the superconducting quantum interferencedevice and the additional superconducting quantum interference deviceare typically each periodic in the magnetic field, and according to oneembodiment a respective magnetic flux input to the inductors of thesuperconducting quantum interference device and the additionalsuperconducting loop are offset equivalent to about one-half of theperiodicity in the device output.

The at least one Josephson junction may be configured as part of atleast one additional superconducting quantum interference device eachhaving a respective additional superconducting quantum interferencedevice output, the at least one additional superconducting quantuminterference device exhibiting a substantially different periodicity inits respective device output with respect to magnetic field than thesuperconducting quantum interference device. The superconducting quantuminterference device output and the at least one respective additionalsuperconducting quantum interference device output may be connected inseries, and the detector output is substantially linear with respect tochanges in magnetic field over a range of magnetic fields outside of anull field, for which neither the superconducting quantum interferencedevice nor the at least one respective additional superconductingquantum interference device is more linear.

The detector may be configured to provide two identical arrays, eacharray comprising at least four superconducting quantum interferencedevices, one of the at least four superconducting quantum interferencedevices being the superconducting quantum interference device, eachsuperconducting quantum interference device having a superconductingloop with a coupling inductance and at least two Josephson junctions,the array having a device output that is periodic in the magnetic fieldinput, and substantially linear in magnetic field over a substantialportion of a quadrant of the field periodicity, connected to provide atleast two of the at least four superconducting quantum interferencedevices respectively in parallel, and at least two of the at least foursuperconducting quantum interference devices respectively in series,each of the identical arrays being provided with a relative magneticfield offset of one-half of the field periodicity with respect to theother; with the detector output representing a differential output fromthe outputs of the two identical arrays. A coupling inductance of atleast one superconducting quantum interference devices may be shunted bya linear resistance. A coupling inductance of at least onesuperconducting quantum interference devices may be shunted by avariable inductor device with an inductance selected to generate asubstantially linear periodic field dependence within a quadrant of thefield periodicity. The variable inductor device may comprise the atleast one Josephson junction. A radio frequency signal may beinductively coupled to the two identical arrays. A magnetic loop antennastructure may be available to provide the radio frequency signal.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary, as well as the following detailed description ofthe embodiments of the present invention, will be better understood whenread in conjunction with the appended drawings. For the purpose ofillustrating the invention, there are shown in the drawings embodimentswhich include those presently preferred. As should be understood,however, the invention is not limited to the precise arrangements andinstrumentalities shown.

FIG. 1 shows basic SQUID structures of the prior art.

FIG. 2 shows an iterative algorithm for successively linearizing thevoltage across a serial array of SQUID cells by adjusting thedistribution of areas of the cells.

FIG. 3 shows a differential array structure consisting of two seriesarrays of SQUIDs with current biasing I_(b)=I_(c); in one array, eachcell is biased by magnetic flux Φ₀/2, and an additional flux bias ofΦ₀/4 is applied to all cells to set the operating point.

FIG. 4 shows a schematic of a bi-SQUID and its periodic voltageresponse, as contrasted with that of a conventional SQUID without theshunt junction.

FIG. 5 shows the model dependence of the linearity (in dB) vs. thecritical current I_(c3) of the shunt junction for a bi-SQUID, forseveral values of inductance L.

FIG. 6 shows the model dependence of the linearity on the output signalamplitude.

FIG. 7 shows the circuit schematic of a bi-SQUID cell for amplifier andantenna applications.

FIG. 8 shows the experimental dependence of voltage on signal current(proportional to flux) for a fabricated bi-S QUID.

FIG. 9A shows the experimental current dependence of an array of 12bi-SQUIDs.

FIG. 9B shows the differential voltage output of two arrays of 12bi-SQUIDs, with magnetic frustration of Φ₀/2 between the two arrays.

FIG. 10A shows the found optimal distribution of the cell areas along aparallel SQIF.

FIG. 10B shows the Voltage response (solid line) of the differentialcircuit of two parallel SQIFs with the found cell area distribution atoptimal magnetic frustration, in which the response linearity within theshaded central area is as high as 101 db, and the frustrated SQIFresponses are shown by dashed lines.

FIGS. 11A and 11B show the linearity of the differential voltageresponse versus both the number N of SQIF cells (FIG. 11A) and thespread in the cell areas at N=36 (FIG. 11B).

FIG. 12 shows a two-dimensional differential serial-parallel SQIFstructure.

FIG. 13A shows the voltage response of a parallel array of N=6 junctionscoupled by inductances with normalized value l=1 at different shuntingresistors R_(sh) connected in parallel to the coupling inductances, inwhich the dashed line shows the voltage response of the array in thelimit of small coupling inductance.

FIG. 13B shows the voltage response of the differential circuit of twofrustrated parallel arrays of N=6 Josephson junctions at l=1 andR_(sh)=R_(N), where R_(N) is the Josephson junction normal resistance.

FIG. 14 shows an active electrically small antenna based on atwo-dimensional differential serial-parallel SQIF-structure.

FIG. 15 shows the dependence of the normalized low-frequency spectraldensity of the resistor voltage noise on number N of resistors R_(N)connected in parallel by coupling inductances at different normalizedvalues l of the inductances.

FIG. 16 shows the dependence of the normalized spectral density of theresistor voltage noise in a parallel array of 30 resistors R_(N) onnormalized frequency at different normalized values l of the couplinginductances.

FIG. 17 shows a normalized transfer factor B=dV/dΦ for a parallel arrayof Josephson junctions versus number of junctions N at differentnormalized values l of coupling inductances.

FIG. 18 shows the normalized voltage response amplitude V_(max) for twoparallel arrays of Josephson junctions coupled correspondingly byunshunted inductances (lower curve) and by the optimally shuntedinductances (upper curve) versus normalized inductance value.

FIG. 19A shows a serial array with stray capacitances and typical I-Vcurves of a serial array of 10 DC SQUIDs calculated using the RSJ modelin the presence of stray capacitances and without capacitances.

FIG. 19B shows the experimentally measured I-V curve of a serial arrayof 20 SQUIDs fabricated using standard niobium integrated circuittechnology.

FIG. 20A shows a schematic of a system subject to stray capacitances inthe SQUID array structures fabricated using standard niobium technologywith two screens (upper and lower screens) in the case of a continuousdouble screen.

FIG. 20B shows a schematic of a first system which reduces straycapacitances in the SQUID array structures fabricated using standardniobium technology with two screens (upper and lower screens) in thecase of an individual double screen.

FIG. 20C shows a schematic of a second system which reduces straycapacitances in the SQUID array structures fabricated using standardniobium technology with two screens (upper and lower screens) in thecase of an intermittent (dashed) double screen.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS I. Bi-SQUID

As described above, a major part of the invention comprises a new SQUIDcell, the bi-SQUID. The DC SQUID, modified by adding a Josephsonjunction shunting the loop inductance, provides extremely high linearitywith the proper selection of parameters. This is somewhat surprising,since a Josephson junction presents a nonlinear inductance. However, thejunction nonlinearity is able to compensate the nonlinearity of thedevice in order to achieve an improved linearity close to 120 dB forsignificant loop inductances (which are necessary to achieve largecoupling to external signals). It is to be understood by those skilledin the art that any other nonlinear reactance that functions in asimilar way would have a similar effect on reducing the nonlinearity ofthe system transfer function.

The linearity dependence of the shunt junction I_(c3) on criticalcurrent at different inductances of the SQUID loop is shown in FIG. 5.The linearity is calculated using a single-tone sinusoidal flux input(of amplitude A/A_(max)=0.2, where A_(max) corresponds to the fluxamplitude Φ₀/4), and measuring the total harmonic distortion in dB. Thisresult shows that the linearity is sharply peaked for each value ofl=LI_(c)/Φ₀, but with different optimized values of I_(c3). Very largevalues of linearity as high as ˜120 dB are achievable. FIG. 6 shows howthe linearity parameter varies as a function of the signal amplitude forother parameters fixed. The linearity decreases as the signal approachesthe maximum value.

A serial array of bi-SQUIDs can be implemented to increase the dynamicrange up to a value comparable with the response linearity. Moreover, aserial SQIF providing a single (non-periodic) voltage response with asingle triangular dip at zero magnetic flux can be implemented.

Single bi-SQUIDs, serial arrays of bi-SQUIDs, and a prototype of anactive electrically small antenna based on a bi-SQUID-array weredesigned, fabricated and tested, using a 4.5 kA/cm² Nb HYPRES process(Hypres Inc., Elmsford N.Y.). The layout design of the chips with theseelements was made before the completion of the numerical simulationsaimed at the optimization of the circuit parameters, in particularbefore obtaining the results presented in FIG. 5 were obtained.Therefore the critical currents of all Josephson junctions in bi-SQUIDswere chosen equal (I_(c1)=I_(c2)=I_(c3)=I_(c)) while the optimal shuntshunting junction critical current should be somewhat less for theimplemented inductance parameter l=1.4.

FIG. 7A shows the schematic equivalent circuit of the bi-SQUID for boththe fabricated single bi-SQUID and the serial array of bi-SQUIDs, foramplifier applications. To apply magnetic flux, a control strip linecoupled magnetically with an additional transformer loop was used. Thecoupling loop with high inductance L_(ex) is connected in parallel toinductance L_(in) and therefore practically does not change theinterferometer inductance. FIG. 7B shows the corresponding equivalentcircuit of the Bi-SQUID for an electrically small antenna.

The voltage response of the bi-SQUID to applied flux (as measured incurrent units) is shown in FIG. 8. The applied bias current was slightlymore than 2I_(c) for the bi-SQUID. The shunt junction critical currentis not optimal at the implemented inductance parameter l=1.4. As aresult, the observed voltage response is not perfectly linear, althoughit shows a clear triangular shape. The measured transfer functionclosely coincides with simulations, however. As for the small hysteresisat the flux value close to ±Φ₀/2, this indicates that effectiveinductance parameter of a single-junction SQUID l*≡l·i_(c3)≡2πLI_(c3)/Φ₀is more than 1 and hence the static phase diagram becomes hysteretic.

The voltage response of the 12-element bi-SQUID array is presented inFIG. 9A, and looks virtually identical to that for a single bi-SQUID.The applied bias current was slightly more than the critical current ofthe array. The voltage response linearity of both bi-SQUID and bi-SQUIDserial array can be further improved by means of differential connectionof two identical bi-SQUIDs or serial arrays oppositely frustrated byhalf a flux quantum. This improvement results from cancellation of alleven harmonics of the individual responses. FIG. 9B shows the voltageresponse of the differential scheme of two serial arrays of 12 bi-SQUIDsfrustrated by half a flux quantum as well as the source responses of thearrays. The arrays are biased about 10% above their critical currents.

II. SQIF-Based Differential Structures

The differential scheme of two parallel SQIFs oppositely frustrated byan applied magnetic field δB (see FIG. 3) is able to provide extremelylinear voltage response in case of a proper choice of the SQIFstructure. In the limit of vanishing inductances l of the interferometercells, one can use an analytical relation for the parallel SQIF response[3]-[5]:V(B)=V _(c)√{square root over ((I _(b) /I _(c))² −|S _(K)(B)|²)}{squareroot over ((I _(b) /I _(c))² −|S _(K)(B)|²)}  (5)

where S_(K)(B) is the structure factor:

$\begin{matrix}{{{S_{K}(B)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{\exp( {{\mathbb{i}}\frac{2\pi}{\Phi_{0}}B{\sum\limits_{m = 1}^{k - 1}a_{m}}} )}}}},} & (6)\end{matrix}$

where I_(b) is the bias current, I_(c) is the total critical current, Kis the number of Josephson junctions, and a_(m) is the area of the m-thinterferometer cell. For sufficiently large K, one can use integrationinstead of summation, and relation (5) can be transformed as follows:

$\begin{matrix}{{S(B)} = {\frac{1}{L}{\int_{0}^{L}\ {{\mathbb{d}z} \cdot {{\exp( {{\mathbb{i}}\frac{2\pi}{\Phi_{0}}B{\int_{0}^{Z}{{a(x)}{\mathbb{d}x}}}} )}.}}}}} & (7)\end{matrix}$

A solution for the specific distribution of the interferometer cellareas a(x) along the SQIF-structure (0<x<L) to make the differentialcircuit voltage responseΔV(B)=V(B+δB)−V(B−δB)  (8)

close to the linear relationΔV(B)=k·B  (9)

in a signal region −α·δB<B<α·δB, where α≦1 is sought.

Relations (5)-(9) allow derivation of master equations and minimizingthe resulting functional to obtain an optimal distribution a(x). One canuse an iterative algorithm to find the problem solution, starting fromsome initial approximation (see FIG. 2). In the case of finiteinductances l of the interferometer cells, the SQIF response V(B) has tobe calculated by means of numerical simulation, using in particular thewell known software PSCAN [18].

The problem can have more than one solution. Various analyticalapproximations for the problem solution at l=0 are found; the best oneis as follows:a(x)/a _(Σ)=1.2−0.48 sin³(πx),  (10)

where a_(Σ)−total area of the parallel SQIF.

FIGS. 10A and 10B show both the cell area distribution (FIG. 10A) (10)and the differential circuit voltage response (FIG. 10B). Linearity ofthe voltage response within the shaded central area is as high as 101dB. To estimate the linearity a sin-like input signal was applied, andthe spectrum of the output signal then studied. A ratio of the basicharmonic to the maximal higher one was used to characterize the responselinearity. It was found that a very high linearity can be obtained usinga relatively small number N of SQIF cells with areas fitted to (10).FIG. 11A shows that the linearity increases rapidly with the number Nand at N>35 reaches a plateau where the linearity is as high as 101 dB.As for the impact of technological spread in the cell areas, FIG. 11Bshows that the tolerable spread is about 4% at N=36; and then thelinearity decreases with the spread value. Approximately the same resultwas obtained for the spread in critical currents of Josephson junctions.A further increase of N can be used to decrease the impact of thetechnological spread in the SQIF circuit parameters as well as toincrease the dynamic range proportional to √{square root over (N)} up tothe linearity level obtained.

Both the dynamic range and the output signal amplitude can beadditionally increased by connection of the differential SQIF structuresin series, i.e., by providing a two-dimensional differentialserial-parallel SQIF structure (see FIG. 12). The number K of theelements connected in series is responsible for the output signalamplitude, while the total number of Josephson junctions N*=N·K isresponsible for the dynamic range of the structure. By varying thenumber of elements connected in parallel (N) and in series (K), one canchange the impedance of the structure over a wide range.

III. Effects of Real Junctions and Coupling Inductances

At the same time, there are several problems which should be solved torealize the potentially high performance of the amplifier or antenna.First of all, one should note that the optimal specific structure of theparallel SQIF reported in [5] was determined based on the ideal RSJmodel of Josephson junctions and for the case of vanishing couplinginductances (L=0). Deviations of junctions and inductors from idealtheoretical behavior will hinder the linearity of the real structurefabricated. There are two general approaches to the problem solution:(i) to provide the closest approach of the experimentalJosephson-junction characteristics to the ones given by the RSJ modeland (ii) to synthesize an optimal SQIF structure founded in experimentalJosephson-junction characteristics by means of numerical simulationtechnique (for example by software PSCAN [18]) and an iterativealgorithm (FIG. 2). Indeed, an optimal strategy may be based on acombination of schemes.

In particular, as for the coupling inductance L, the negative influenceof the finite value of L on the voltage response linearity can bereduced by shunting resistors R_(SH) connected in parallel to theinductances. Due to the fact that the impedance of the RL circuitbecomes low enough at the Josephson oscillation frequency, the parallelarray voltage response approaches that for smaller and smallerinductance with the decrease of R_(SH) down to some optimal resistancevalue depending on the normalized inductance l; further increase inR_(SH) leads to some other linearity distortions. Therefore, the mosteffective method is synthesis of an optimal SQIF structure with the cellarea distribution a(x) optimized for the finite value of l. In this caseone should use a high performance numerical simulation technique (e.g.,PSCAN software [18]) for calculation of the SQIF voltage response V(Φ)in every cycle of the iterative algorithm (FIG. 2), which has to be usedto solve the master equation.

The shunting technique efficiency is confirmed by results of numericalsimulations presented in FIGS. 13A and 13B. One can see that atR_(SH)≈0.1R_(N) (where R_(N) is Josephson junction normal resistance),the voltage response of the parallel array of 6 junctions withl≡2πI_(C)L/Φ₀=1 approaches that for vanishing coupling inductances. As aconsequence, the required linear voltage response of the differentialscheme of two parallel SQIFs with N=20 and coupling inductances l=1 eachshunted by resistor R_(SH)=0.1R_(N) are observed.

IV. Advantages of SQIF-Like Structures

In the case of a serial SQIF including N DC SQUIDs, the thermal noisevoltage V_(F) across the serial structure is proportional to square rootof N, while the voltage response amplitude V_(max)(Φ) and the transferfactor B=∂V/∂Φ both are about proportional to N^(1/2). This means thatthe dynamic range D=V_(max)(Φ)/V_(F) increases as N^(1/2). As for theparallel SQIF, in the case of vanishing coupling inductances (l=0), thedynamic range is also proportional to square root of number of junctionsN. In fact, the thermal noise voltage V_(F) across the parallelstructure decreases with the square root of N, while the voltageresponse amplitude V_(max)(Φ) remains constant and the transfer factorB=∂V/∂Φ increases as about N.

A SQIF-like structure is characterized by a superior broadband frequencyresponse from DC up to approximately 0.1·ω_(c), where ω_(c) ischaracteristic Josephson frequency [13]. Therefore, a further increasein characteristic voltage V_(c) of Josephson junctions by implementationin niobium technology with higher critical current density, or by use ofhigh-T_(c) superconductors, promises an extension of the frequency bandup to several tens of gigahertz. Moreover, the SQIF eliminates highinterference, and it sufficiently decreases the well known saturationproblem of SQUID-based systems. Therefore, SQIF-based systems can easilyoperate in a normal lab environment.

An approach to synthesis of multi-SQUID serial structures has beenreported, capable of providing periodic high linearity voltage response[11, 12]. The approach is based on the formation of serial structureswhich are capable of providing periodic triangular voltage response to amagnetic field B. Using interferometer cells with a harmonic voltageresponse, one can synthesize a serial array consisting of many groups ofidentical interferometers, each group providing a specific spectralcomponent of the resulting voltage response of the array. According toestimations, the response linearity reaches 120 dB, if the number of thegroups is as high as about 165. The second way to synthesize a highlylinearity array structure is through implementation of a differentialscheme of two serial arrays of DC interferometers biased by currentI_(b)=I_(C) (critical current biasing), where I_(C) is theinterferometer critical current.

According to an embodiment, a more advanced system is providedcomprising one- and two-dimensional multi-element structurescharacterized by SQIF-like high linearity voltage response. Thestructures are based on use of a differential scheme of two magneticallyfrustrated parallel SQIFs, with both a specific cell area distributiona(x) along array and a critical current biasing (see FIG. 3).Optimization of the cell area distribution allows an increase of thevoltage response linearity up to the levels required. This optimizationcan be performed numerically by solution of a master equation with theaid of an iterative algorithm.

A multi-element structure synthesized according to the presentembodiments can be used, for example, to provide high performanceamplifiers. The proposed two-dimensional structure can also used as anactive antenna device. The efficiency of the antenna can besignificantly increased by combining it with a reflecting parabolicantenna. By varying the number of elements connected in parallel (N) andin series (K), one can set the impedance to a value needed to optimallymatch the antenna load used.

The high expectation for the multi-element SQIF-like structures is basedon estimations based on idealized structures, as well as on the voltageresponse characteristics calculated with use of RSJ model. However, thetrue characteristics of the actually realized array structures may bedifferent. Limitations imposed by finite coupling inductances and straycapacitances are discussed below.

The finite value of coupling inductances l between Josephson junctionsin a parallel array is of importance for all principal characteristicsof the array, because of limitations on the coupling radius.

The finite coupling radius limits an increase of both the dynamic rangeand the transfer factor dV/dΦ with increase of number of junctions N. Tostudy the noise characteristics in a clearer and more powerful manner,one can perform numerical simulation of a parallel array of theinductively coupled resistors R_(N), each connected to an individualsource of white-noise current.

FIG. 14 shows an active electrically small antenna based ontwo-dimensional differential serial-parallel SQIF-structure (the filledstructure in central part of chip). SQIF sections are connected bystrips of normal metal. The chip contains a regular matrix of identicalblocks of parallel SQIFs, to provide a homogeneous magnetic fielddistribution in the center part of the chip. The inset shows such ablock with a parallel SQIF. The shown SQIF is topology-oriented for ahigh-Tc superconductor technology.

FIG. 15 shows the dependence of the low-frequency spectral densityS_(v)(0) of the resistor voltage noise on the number of resistors N atdifferent values of normalized coupling inductance l. The data arepresented for normalized frequency ω/ω_(c)=10⁻³ corresponding closely tothe signal frequency range in a SQUID/SQIF amplifier (here ω_(c) ischaracteristic Josephson frequency). Within the coupling radius, thespectral density S_(v)(0) decreases as 1/N and then it comes to aconstant value when number N becomes more than coupling radius dependingon l.

FIG. 16 shows the spectral density S_(v)(ω) versus normalized frequencyranged from 0.01 to 1 for parallel array of 30 resistors R_(N). At bothcoupling inductances l=3 and l=1, the spectral density S_(v)(ω)monotonically increases with frequency and remains constant at l=0.001.It reflects a decrease in coupling radius with frequency for bothinductances l=3 and l=1, as well as the fact that the coupling radius atl=0.01 exceeds the size of the array of 30 elements in the entirefrequency range.

One can see that implementation of noiseless resistors R_(SH)=0.1R_(N)shunting the inductances l=1 stops both the coupling radius decrease andthe noise spectral density increase at ω/ω_(c)≧0.1 (see dashed line inFIG. 16). A proper account of the respective noises of the shuntingresistors will lift the curve about two times as high, as if thespectral density for the noise current sources connected to basicresistors R_(N) becomes more by factor k≈4-5.

FIG. 17 shows the dependence of the normalized transfer factor B=dV/dΦfor a parallel array of Josephson junctions versus number of junctions Nat different normalized values l of coupling inductances. The dashedcurve shows the transfer factor dependence for l=1 when all the couplinginductances are shunted by resistors R_(SH)=R_(N). The observedsaturation in the transfer factor, depending on N, is reached whennumber of junctions exceeds the coupling radius at frequency ω/ω_(c)˜1.

In such a way, increases in dynamic range D=V_(max)(Φ)/V_(F) with thenumber N of Josephson junctions in a parallel array are limited by thecoupling radius at finite coupling inductances. Shunting of theinductances for improving linearity of the differential SQIF voltageresponse does not really change the dynamic range. In fact, the observedincrease in the voltage response amplitude V_(max)(Φ) (see FIG. 18) iscompensated by an increase in V_(F) owing to the noise of the shunts.

In the case of an unloaded serial array of DC SQUIDs, the dynamic rangedoes actually increase with the number N of interferometer cells.Nevertheless, in reality, stray capacitances and load impedance are bothable to substantially change the I-V curve of the array, and hence theamplitude V_(max) and form of the array voltage response. The decreasein V_(max) leads to a proportional decrease in dynamic range. The changein the voltage response curve reduces linearity of the whole arraystructure.

FIGS. 19A and 19B shows the typical impact of the stray capacitances onI-V curve of the serial array of DC SQUIDs. The contribution of thestray capacitance of each SQUID increases with the SQUID position fromground to signal terminal. Stray capacitances cause the I-V curve toappear similar to a hysteresis curve, as well as to form one or evenmore undesired features on the I-V curve. The features shown result froma phase-locking phenomenon. In the solid line labeled a of FIG. 19B, thefeatures of the I-V curves of the array cells do not coincide because ofdifferent “capacitive loads.” In the dashed line labeled b, the featuresof all the individual I-V curves coincide because of mutualphase-locking of the Josephson-junction oscillations.

The fabrication of serial arrays based on standard niobium technologyusing two superconducting screens is accompanied by undesirably highstray capacitances (see FIG. 20A). To essentially decrease the impact ofthe capacitance, individual double screening may be used for each SQUIDas shown in FIG. 20B and FIG. 20C. Both schemes are characterized by theI-V curve b in FIG. 13B, but the latter one provides lower inductancesof the strips which connect the SQUID cells.

V. Conclusion

Advantages of one- and two-dimensional SQIF-like structures formicrowave applications as high-performance amplifying devices arereadily apparent from their ability to provide an increase in dynamicrange with a number of elements as well as high linearity when employinga properly specified array structure. Linearity can be especiallyenhanced using cells comprising the bi-SQUID structure. At the sametime, there are some fundamental limitations imposed by finite couplinginductances, stray capacitances and parasitic couplings. Therefore,implementation of high-performance devices preferably employs carefuland detailed analysis of the multi-element array structure, taking intoconsideration all the true parameters including all parasitic parametersand couplings. A differential scheme comprising two magneticallyfrustrated parallel SQIFs is developed to obtain a highly linearsingle-peak voltage response. The response linearity can be increased upto 120 dB by means of a set of properly specified cell area distributionof the SQIFs. The high linearity is attainable with a relatively smallnumber of junctions. Such a circuit provides a high-performancetwo-dimensional serial-parallel SQIF-like array. Varying the number ofelements connected in parallel, and in series, permits setting theimpedance value needed to solve the problem related to negative impactof the load used. The synthesized structures can be used to designhigh-efficiency amplifiers and electrically small active antennae foruse in the gigahertz frequency range. The efficiency of the antenna canbe significantly increased by combination with a reflecting parabolicantenna.

It should be appreciated that changes could be made to the embodimentsdescribed above without departing from the inventive concepts thereof.It should be understood, therefore, that this invention is not limitedto the particular embodiments disclosed, but it is intended to covermodifications within the spirit and scope of the present invention asdefined by the appended claims.

REFERENCES

Each of the following is expressly incorporated herein by reference:

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What is claimed is:
 1. A superconducting radio frequency amplifier,comprising a plurality of Josephson junctions, each having a respectiveoutput, wherein a radio frequency input is distributed inductively toeach of the plurality of Josephson junctions, and the respective outputsof the plurality of Josephson junctions are combined into a compositesignal having a total harmonic distortion for at least one sinusoidalinput signal of less than −80 dB.
 2. The amplifier of claim 1, whereinthe composite signal has a total harmonic distortion for at least onesinusoidal input having a frequency above 1 GHz signal of less than −80dB, and wherein a power of the composite signal is greater than a powerof the radio frequency input.
 3. The amplifier of claim 1, whereby theplurality of Josephson junctions are integrated with a plurality ofantenna structures, so as to form an active radio frequency antennareceiver array.
 4. The amplifier of claim 1, whereby the plurality ofJosephson junctions comprise niobium.
 5. The amplifier of claim 1,whereby the plurality of Josephson junctions comprise a high-Tcsuperconductor.
 6. The amplifier of claim 1, further comprising at leastone superconductor screen configured to reduce a stray capacitanceeffect.
 7. The amplifier of claim 1, wherein the plurality of Josephsonjunctions are fabricated on a chip using integrated circuit technology.8. The amplifier of claim 1, wherein the plurality of Josephsonjunctions comprise a plurality of Superconducting Quantum InterferenceDevices (SQUIDs) configured in an array.
 9. The amplifier of claim 8,wherein the array comprises a plurality of SQUIDs connected in series.10. The amplifier of claim 9, wherein each SQUID has a loop having aloop area, wherein the distribution of the effective loop areas of theSQUIDs connected in series are defined according to an iterativealgorithm which optimizes the linearity of the amplifier output.
 11. Theamplifier of claim 9, wherein two identical series-connected arrays ofSQUIDs are biased in a differential manner so as to cancel out evenharmonic distortion in a combined output of the two identical seriesconnected arrays of SQUIDs.
 12. The amplifier of claim 1, wherein theplurality of Josephson junctions comprise a plurality of SuperconductingQuantum Interference Filters (SQIFs) configured in an array.
 13. Theamplifier of claim 1, where the plurality of Josephson junctions areconfigured as at least one Bi-SQUID comprising a SQUID shunted by aJosephson junction with parameters optimized so as to generate atriangle-wave output as a function of linear input.
 14. A method ofamplifying a radio frequency signal, comprising: providing a pluralityof superconducting components, each exhibiting an intrinsic harmonicdistortion, configured in an array so as to present an amplified radiofrequency output that compensates the intrinsic harmonic distortion inthe configured plurality of superconducting components, having alinearity of at least 80 dB with respect to a radio frequency input;providing a radio frequency input; and generating the amplified radiofrequency output.
 15. The method of claim 14, wherein the plurality ofsuperconducting components each comprise a Superconducting QuantumInterference Device (SQUID).
 16. The method of claim 15, wherein thearray comprises a plurality of SQUIDs connected in series.
 17. Themethod of claim 16, wherein each SQUID has a loop having a loop area,wherein the distribution of the effective loop areas of the SQUIDsconnected in series are defined according to an iterative algorithmwhich optimizes the linearity of the amplifier output.
 18. The method ofclaim 16, wherein two identical series connected arrays of SQUIDs arebiased in a differential manner so as to cancel out even harmonicdistortion in a combined output of the two identical series connectedarrays of SQUIDs.
 19. The method of claim 14, wherein the plurality ofsuperconducting components each comprise a Superconducting QuantumInterference Filter (SQIF).
 20. The method of claim 14, where theplurality of superconducting components comprise at least one Bi-SQUIDcomprising a SQUID shunted by a Josephson junction with parametersoptimized so as to generate a triangle-wave output as a function oflinear input.